Clearly, if \(X,A \in Obj\; \mathcal {C}\)  and \(X<^s A\) , then \(X<^p A\)  but the converse does not hold in general. For example, in the category of groups, there exists a countable torsion-free abelian group \(A\)  such that \(A\cong A\oplus A\oplus A\)  but \(A\lnot \cong A\oplus A\)  (see [1]}). It follows that \(A\leqslant ^d A\oplus A\)  and \(A\oplus A \leqslant ^d A\)  but \(A\lnot \cong A\oplus A\) . Therefore, \(A<^p A\oplus A\)  but \(A\lnot <^s A\oplus A\) . Also, in the homotopy category of CW-complexes and homotopy classes of cellular maps between them, there exist two CW-complexes \(X\)  and \(Y\)  for which \(X\leqslant ^d Y\)  and \(Y\leqslant ^d X\)  but \(X\lnot \simeq Y\)  (see [2]}). Accordingly, \(X<^p Y\)  but \(X\lnot <^s Y\) .

The capacity \(C(A)\)  of an \(A \in Obj\; \mathcal {C}\)  is the cardinality of the class of isomorphism classes of all the \(X \in Obj\; \mathcal {C}\)  such that \(X\leqslant ^d A\) . Also, the strong capacity \(SC(A)\)  of an \(A \in Obj\; \mathcal {C}\)  is the cardinality of the class of isomorphism classes of all \(X \in Obj\; \mathcal {C}\)  such that \(X<^s A\) . If there is no such \(X\) , we write \(SC(A)=0\) .

The previous definition of the capacity of a compatum in the shape category of compacta coincides with Borsuk's definiton of the capacity (see [3]}).
